| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl1 1192 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
𝐴 ∈
ℂ) | 
| 2 |  | simpl3 1194 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
𝑈: ℋ⟶
ℋ) | 
| 3 |  | simpr 484 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
𝑥 ∈
ℋ) | 
| 4 |  | homval 31760 | . . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
((𝐴
·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥))) | 
| 5 | 1, 2, 3, 4 | syl3anc 1373 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝐴
·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥))) | 
| 6 | 5 | fveq2d 6910 | . . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝑇‘((𝐴 ·op
𝑈)‘𝑥)) = (𝑇‘(𝐴 ·ℎ (𝑈‘𝑥)))) | 
| 7 |  | homulcl 31778 | . . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ)
→ (𝐴
·op 𝑈): ℋ⟶ ℋ) | 
| 8 | 7 | 3adant2 1132 | . . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝐴
·op 𝑈): ℋ⟶ ℋ) | 
| 9 |  | fvco3 7008 | . . . . 5
⊢ (((𝐴 ·op
𝑈): ℋ⟶ ℋ
∧ 𝑥 ∈ ℋ)
→ ((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = (𝑇‘((𝐴 ·op 𝑈)‘𝑥))) | 
| 10 | 8, 9 | sylan 580 | . . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = (𝑇‘((𝐴 ·op 𝑈)‘𝑥))) | 
| 11 |  | fvco3 7008 | . . . . . . 7
⊢ ((𝑈: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥))) | 
| 12 | 2, 3, 11 | syl2anc 584 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥))) | 
| 13 | 12 | oveq2d 7447 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝐴
·ℎ ((𝑇 ∘ 𝑈)‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) | 
| 14 |  | lnopf 31878 | . . . . . . . . 9
⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶
ℋ) | 
| 15 | 14 | 3ad2ant2 1135 | . . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ 𝑇: ℋ⟶
ℋ) | 
| 16 |  | simp3 1139 | . . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ 𝑈: ℋ⟶
ℋ) | 
| 17 |  | fco 6760 | . . . . . . . 8
⊢ ((𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ 𝑈): ℋ⟶
ℋ) | 
| 18 | 15, 16, 17 | syl2anc 584 | . . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ 𝑈): ℋ⟶
ℋ) | 
| 19 | 18 | adantr 480 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝑇 ∘ 𝑈): ℋ⟶
ℋ) | 
| 20 |  | homval 31760 | . . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op
(𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))) | 
| 21 | 1, 19, 3, 20 | syl3anc 1373 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝐴
·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))) | 
| 22 |  | simpl2 1193 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
𝑇 ∈
LinOp) | 
| 23 | 16 | ffvelcdmda 7104 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝑈‘𝑥) ∈ ℋ) | 
| 24 |  | lnopmul 31986 | . . . . . 6
⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ (𝑈‘𝑥) ∈ ℋ) → (𝑇‘(𝐴 ·ℎ (𝑈‘𝑥))) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) | 
| 25 | 22, 1, 23, 24 | syl3anc 1373 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝑇‘(𝐴 ·ℎ (𝑈‘𝑥))) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) | 
| 26 | 13, 21, 25 | 3eqtr4d 2787 | . . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝐴
·op (𝑇 ∘ 𝑈))‘𝑥) = (𝑇‘(𝐴 ·ℎ (𝑈‘𝑥)))) | 
| 27 | 6, 10, 26 | 3eqtr4d 2787 | . . 3
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥)) | 
| 28 | 27 | ralrimiva 3146 | . 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ ∀𝑥 ∈
ℋ ((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥)) | 
| 29 |  | fco 6760 | . . . 4
⊢ ((𝑇: ℋ⟶ ℋ ∧
(𝐴
·op 𝑈): ℋ⟶ ℋ) → (𝑇 ∘ (𝐴 ·op 𝑈)): ℋ⟶
ℋ) | 
| 30 | 15, 8, 29 | syl2anc 584 | . . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ (𝐴 ·op
𝑈)): ℋ⟶
ℋ) | 
| 31 |  | simp1 1137 | . . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ 𝐴 ∈
ℂ) | 
| 32 |  | homulcl 31778 | . . . 4
⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ) → (𝐴 ·op
(𝑇 ∘ 𝑈)): ℋ⟶
ℋ) | 
| 33 | 31, 18, 32 | syl2anc 584 | . . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝐴
·op (𝑇 ∘ 𝑈)): ℋ⟶
ℋ) | 
| 34 |  | hoeq 31779 | . . 3
⊢ (((𝑇 ∘ (𝐴 ·op 𝑈)): ℋ⟶ ℋ
∧ (𝐴
·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ) →
(∀𝑥 ∈ ℋ
((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) ↔ (𝑇 ∘ (𝐴 ·op 𝑈)) = (𝐴 ·op (𝑇 ∘ 𝑈)))) | 
| 35 | 30, 33, 34 | syl2anc 584 | . 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (∀𝑥 ∈
ℋ ((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) ↔ (𝑇 ∘ (𝐴 ·op 𝑈)) = (𝐴 ·op (𝑇 ∘ 𝑈)))) | 
| 36 | 28, 35 | mpbid 232 | 1
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ (𝐴 ·op
𝑈)) = (𝐴 ·op (𝑇 ∘ 𝑈))) |